Paragraph: Let (8+3 7 )^ 20 =U+V and (8-3 7 )^ 20 =W , where U is an integer and 0 Question: What is V+W equal to?

Correct answer: D. 1 We have, (8+3 7 )^ 20 =U+V ....(i) and (8-3 7 )^ 20 =W ....(ii) Here, o < W < 1 Now, adding equation (i) \& (ii), we have, aligned & U+V+W=(8+3 7 )^ 20 +(8-3 7 )^ 20 & =2[ ^ 20 C_0 · 8^ 20 + ^ 20 C_2 8^ 18 ·(3 7 )^2+ .+(3 7 )^ 20 ] aligned R.H.S. is an even number. Also, o < V < 1, o < W < 1 and U is an integer V+W is an integer V+W=1 So, V+W=1

NDA 2024

Exam: NDA
Year: 2024
Subject: Mathematics
Chapter: Binomial Theorem

NDA 2024 · Mathematics · Binomial Theorem

Paragraph: Let (8+3 7 )^ 20 =U+V and (8-3 7 )^ 20 =W , where U is...

Paragraph: Let

(8+3 \sqrt{7})^{20}=\mathrm{U}+\mathrm{V}

and

(8-3 \sqrt{7})^{20}=\mathrm{W}

, where U is an integer and

0 < \mathrm{V} < 1

.
Question: What is

\mathrm{V}+\mathrm{W}

equal to?

Options

A $8$
B $4$
C $2$
D $1$

Correct answer

D. $1$

Step-by-step solution

We have,

(8+3 \sqrt{7})^{20}=\mathrm{U}+\mathrm{V}

....(i)

and (8-3 \sqrt{7})^{20}=\mathrm{W}

....(ii) Here,

\mathrm{o} < \mathrm{W} < 1

Now, adding equation (i) \& (ii), we have,

\begin{aligned} & \mathrm{U}+\mathrm{V}+\mathrm{W}=(8+3 \sqrt{7})^{20}+(8-3 \sqrt{7})^{20} \\ & =2\left[{ }^{20} \mathrm{C}_0 \cdot 8^{20}+{ }^{20} \mathrm{C}_2 8^{18} \cdot(3 \sqrt{7})^2+\ldots .+(3 \sqrt{7})^{20}\right] \end{aligned}

R.H.S. is an even number. Also,

\mathrm{o} < \mathrm{V} < 1, \mathrm{o} < \mathrm{W} < 1

and U is an integer

\Rightarrow \mathrm{V}+\mathrm{W}

is an integer

\Rightarrow \mathrm{V}+\mathrm{W}=1

So,

V+W=1

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